User blog:Kite in rainbow/A weak and clear version of my Bracket-Comma notation ver2,it reaches Γ(0)
My last notation could beyond ψ(I),but it's too confusion,both growth level and definition. I tried to get a better definition which could be more clear. To get more attention,I want to introduce a weak but easy to understand subset of my notation v2 -- without using the symbol comma. It's much more clear and easier to understand,so it's too weak that it can just reach the level Γ_0. (Sometimes we will regard BS or LBS as an adjective.) 1.We define a set BS (means bracket-string): (1)Empty string φ is in BS. (2)If X(1),X(2) is in BS,then X(1)X(2) and X(1) (so X(2)) is in BS. Now every BS X can be written in the form X(1)X(2)X(3)...X(n). For BS X,we define n(X)=n,and X(i) as we have written. For a closed interval a,b,0≤a,b≤n,define X(a,b)=X(a)X(a+1)...X(b).If b>: N*LBS -> LBS and a map S: LBS -> Power(BS). These three objects must satisfied that: (1)φ is LBS.n>>φ=φ for any n belongs to N. (2)If X is LBS,then X[] is in LBS,n>>X[]=X for any n belongs to N. (3)If X is LBS,S(0) is the set {X},S(i) is the set N>>S(i-1) for positive integer i,then S(X)=union of S(i),i=0,1,2,3,... (4)For any BS X,satisfy these below three situation: ((1))X≠φ,n=n(X),every X(i) is LBS. ((2))The biggest i s.t. X(n) doesn't belong to S(X(i)) exists and satisfy 0>>X(n) belongs to S(X(i)). ((3))Define m to be the biggest number s.t. m>>X(n) belongs to S(X(i)).If m is infinity,set m=1 instead of x.(In fact,you can choose any positive integer.) Define Y(0)=X(1,i-1),Y(1)=X(1,n-1),Y(i+1)=Y(i)m+i>>X(n)X(i+1,n-1) for any positive integer i. For any i belongs to N we have Y(i) is LBS. We have X is BCS and i>>X=Y(i) for any i belongs to N. We define LBS to be the smallest set which can satisfy these four claim with two map >> and S. 3.We define a function -> : N*LBS -> LBS For any n belongs to N, (1) n->φ=n (2) n->X=(n+1)->(n>>X),for any LBS X≠φ. 4.We define a function KIRBN:N->N+ We define a LBS sequence An:A0=[],An+1=[]A(n). And KIRBN(n)=3->A(n). KIRBN has growth rate Γ_0. If the symbol comma can be used,this notation can have growth rate much more than ψ(I),but the definition is complex. 5.Some examples. n>>[][[]][]=[][[]] 3>>[][[]]=[][][] (Some space added to watch it more easily) 4>>3>>[] [ [] ] [ [] [] ]=4>>[] [ [] ] [ [] ] [ [] ]=[] [ [] ] [ [] ] [] [ [] ] [ [] ] [] [ [] ] [ [] ] [] [ [] ] [ [] ] 3>>[] [ [] [ [] ] ]=[] [ [] ] [ [] [] ] [ [] [] [] ] 3>>[] [ [] ] [ [] [] ] [ [] [] [] ]=[] [ [] ] [ [] [] ] [ [] [] ] [ [] [] ] 3>>4>>[] [ [] [ [] [ [] ] ] ]=3>>[] [ [] ] [ [] [ [] ] ] [ [] [ [] ] [ [] [] ] ] [ [] [ [] ] [ [] [] ] [ [] [] [] ] ] =[] [ [] ] [ [] [ [] ] ] [ [] [ [] ] [ [] [] ] ] [ [] [ [] ] [ [] [] ] [ [] [] ] ] [ [] [ [] ] [ [] [] ] [ [] [] ] [ [] [] ] ] Category:Blog posts